
# q-expansion of newform 429.2.i.f, downloaded from the LMFDB on 24 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 429
weight = 2
poly_data = [1, -5, 26, -7, 30, 3, 32, 4, 7, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[31405, -1643, 8480, -159430, 8533, -265, 16404, -583, 318, 911], 160271], [[319366, -149916, 773760, 122097, 778596, -24180, 154140, -53196, 29016, -4571], 160271], [[-672224, 187519, -967840, 52190, -973889, 30245, -199961, 66539, -36294, 4889], 160271], [[388420, -66991, 345760, 288305, 347921, -10805, 9622, -23771, 12966, -5191], 160271], [[155382, -808050, 60405, -933617, -94480, -988556, -126203, -219517, 32316, -31405], 160271], [[-15562, -1431066, -1829, -1969946, -984502, -2138559, -433019, -409567, 33989, -62396], 160271], [[373160, -1943642, 472311, -2305835, -287026, -2774426, -394293, -622469, 91837, -91217], 160271], [[645973, -3358996, 1061492, -3874513, -386333, -4140116, -514881, -902380, 127955, -128866], 160271]]

hecke_ring_character_values = [[287, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [79, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [67, [0, 0, 0, 0, 0, 0, -1, 0, 0, 0]]]
aps_data = [[1, -1, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, -1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 1, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 1, -1, 0, 1, -1], [0, 1, -1, 1, 0, 0, -2, 1, 0, 2], [0, -1, 1, -1, 0, 0, 2, -1, 0, 0], [1, 0, 0, 0, 2, 1, -1, -1, -1, 2], [-2, 0, 0, 0, 0, -1, 2, -2, 1, 0], [0, 0, -2, 1, -2, -3, 0, 0, 0, 0], [4, -1, 0, 0, -1, -1, -4, 0, 1, -1], [0, -2, 0, 0, 1, 3, 0, -1, -3, 1], [0, 0, 0, -4, 0, 0, -1, -4, 0, 1], [-2, 0, 6, -4, -1, -2, 0, 0, 0, 0], [-2, 0, 1, -3, -1, 1, 0, 0, 0, 0], [0, 1, -1, 1, 0, 0, 1, 1, 1, -3], [0, 1, -1, 2, 0, 0, -1, 2, -1, -1], [0, 1, 0, 0, -2, -2, 0, -4, 2, -2], [0, -3, 3, -5, 0, 0, 4, -5, 3, 0], [2, 0, 2, -2, -2, -5, 0, 0, 0, 0], [0, 0, -2, -2, 1, -1, 0, 0, 0, 0], [-1, 0, -7, 1, 2, 0, 0, 0, 0, 0], [5, 1, 0, 0, 0, 0, -5, -3, 0, 0], [0, 1, -1, -4, 0, 0, 2, -4, -3, 4], [1, 3, 0, 0, 1, 1, -1, 5, -1, 1], [2, 0, 4, -4, -2, -5, 0, 0, 0, 0], [-4, 4, 0, 0, 1, 1, 4, 1, -1, 1], [1, 0, 0, -6, -5, 0, 0, 0, 0, 0], [0, 4, -4, 1, 0, 0, -3, 1, 0, -1], [8, -2, 0, 0, -3, -1, -8, 5, 1, -3], [-8, 0, 1, 3, -1, 3, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 6, -3, -4, 1], [0, -5, 5, 3, 0, 0, -4, 3, -2, -3], [0, -5, 5, 1, 0, 0, 4, 1, -3, 6], [-1, 0, -11, 5, -1, 2, 0, 0, 0, 0], [-10, 0, 0, 5, 7, 3, 0, 0, 0, 0], [0, -1, 1, -7, 0, 0, 5, -7, 2, 1], [7, 1, 0, 0, 8, 1, -7, -2, -1, 8], [0, 3, -3, 0, 0, 0, 4, 0, -1, -3], [-3, -4, 0, 0, -3, 1, 3, -1, -1, -3], [-4, 0, 3, 8, -2, 0, 0, 0, 0, 0], [0, -4, 4, -1, 0, 0, -3, -1, -2, 1], [0, -3, 0, 0, -5, 0, 0, 3, 0, -5], [7, -6, 0, 0, 1, 2, -7, -9, -2, 1], [0, -6, 6, 0, 0, 0, 3, 0, -1, -7], [-11, 0, 0, 0, 5, 4, 11, 1, -4, 5], [6, 5, 0, 0, 1, -10, -6, 5, 10, 1], [0, 2, -2, 2, 0, 0, -1, 2, 9, -3], [1, 0, 4, -3, 3, 1, 0, 0, 0, 0], [-2, 0, 0, -4, 0, 5, 0, 0, 0, 0], [-9, 0, -1, 0, -3, 1, 0, 0, 0, 0], [0, 8, -8, 2, 0, 0, 7, 2, -4, 5], [0, -5, 5, -2, 0, 0, 5, -2, 3, 5], [3, 3, 0, 0, -2, -1, -3, -5, 1, -2], [10, -3, 0, 0, 0, -2, -10, 6, 2, 0], [0, -2, 2, -7, 0, 0, 5, -7, -6, 8], [3, -5, 0, 0, -7, -9, -3, 2, 9, -7], [0, 8, -8, 5, 0, 0, -5, 5, 8, 2], [10, 0, -4, 6, 9, 3, 0, 0, 0, 0], [3, 0, 0, 0, -4, 1, -3, 7, -1, -4], [0, 1, -1, -3, 0, 0, 10, -3, -1, 5], [-4, 0, 7, 1, 10, 4, 0, 0, 0, 0], [3, 0, -14, 4, 1, 5, 0, 0, 0, 0], [-16, 0, -5, -2, 6, 8, 0, 0, 0, 0], [-2, 0, -8, 2, 5, 0, 0, 0, 0, 0], [0, 6, -6, 2, 0, 0, -1, 2, 0, 5], [13, 0, 6, -7, -4, 4, 0, 0, 0, 0], [0, 2, -2, 1, 0, 0, 0, 1, 7, -2], [-4, 14, 0, 0, -1, -4, 4, 2, 4, -1], [4, -1, 0, 0, -3, 1, -4, 12, -1, -3], [-17, 0, -7, 1, -2, 3, 0, 0, 0, 0], [11, -6, 0, 0, -1, -6, -11, 5, 6, -1], [0, 7, -7, 1, 0, 0, -11, 1, 6, -3], [15, -1, 0, 0, 8, 0, -15, -6, 0, 8], [0, -3, 3, 9, 0, 0, -14, 9, 5, -4], [9, 0, 3, -9, -10, -13, 0, 0, 0, 0], [0, 6, -6, -7, 0, 0, 6, -7, 7, 2], [-1, 6, 0, 0, 0, -3, 1, -8, 3, 0], [0, -6, 6, 9, 0, 0, -17, 9, 1, -7], [0, -14, 0, 0, 0, 4, 0, -5, -4, 0], [5, 0, 3, 1, 5, 7, 0, 0, 0, 0], [14, -3, 0, 0, -5, 1, -14, 7, -1, -5], [0, 12, -12, 8, 0, 0, -5, 8, 3, -8], [-2, 2, 0, 0, -6, -6, 2, 4, 6, -6], [6, 0, 15, -9, -1, -8, 0, 0, 0, 0], [0, 12, -12, 4, 0, 0, 9, 4, 4, 1], [-4, -6, 0, 0, 3, 0, 4, 0, 0, 3], [0, 0, 0, -2, 0, 0, 3, -2, -9, 0], [6, 0, 12, -7, -8, -6, 0, 0, 0, 0], [3, 0, 9, 2, 9, 4, 0, 0, 0, 0], [3, 11, 0, 0, -2, -10, -3, -2, 10, -2], [0, -6, 6, -13, 0, 0, -3, -13, 2, 0], [-15, -1, 0, 0, 16, 12, 15, -5, -12, 16], [4, 0, 2, 3, 0, 0, 0, 0, 0, 0], [0, 7, -7, 0, 0, 0, -1, 0, 0, -1], [6, -19, 0, 0, -1, 7, -6, -4, -7, -1], [-14, 0, 3, -1, -1, -11, 0, 0, 0, 0], [-12, 8, 0, 0, -4, 2, 12, -7, -2, -4], [-7, 0, -6, -2, 0, 10, 0, 0, 0, 0], [-6, 0, -13, 12, -1, 7, 0, 0, 0, 0], [10, 10, 0, 0, 6, 0, -10, 0, 0, 6], [0, 1, -1, -2, 0, 0, 18, -2, -3, 5], [16, 10, 0, 0, -4, -3, -16, 3, 3, -4], [-3, 0, -1, 9, -3, 2, 0, 0, 0, 0], [12, 0, 2, -3, -4, 9, 0, 0, 0, 0], [4, -8, 0, 0, -6, 0, -4, -3, 0, -6], [2, 0, 7, -4, 2, -8, 0, 0, 0, 0], [8, 0, 12, -4, 5, -7, 0, 0, 0, 0], [-1, -8, 0, 0, 3, 5, 1, -4, -5, 3], [0, 1, -1, -12, 0, 0, 20, -12, -5, 12], [-10, -8, 0, 0, 12, 4, 10, -8, -4, 12], [0, 5, -5, -1, 0, 0, 10, -1, 8, 0], [23, 0, -10, -12, -9, -1, 0, 0, 0, 0], [0, 5, -5, 4, 0, 0, -2, 4, -7, 2], [0, -6, 6, -19, 0, 0, 1, -19, -9, 6], [0, 1, -1, 2, 0, 0, 2, 2, -10, 5], [-15, 5, 0, 0, 2, -9, 15, 3, 9, 2], [28, -3, 0, 0, -12, -15, -28, 8, 15, -12], [0, 11, -11, 5, 0, 0, -15, 5, 7, 3], [7, 3, 0, 0, 12, 9, -7, 0, -9, 12], [0, -8, 0, 0, 2, 9, 0, -6, -9, 2], [-27, 0, 5, 13, 4, 3, 0, 0, 0, 0], [0, 5, -5, 0, 0, 0, 8, 0, -10, 7], [-10, 13, 0, 0, -7, 0, 10, -5, 0, -7], [10, 0, 5, -7, -9, -7, 0, 0, 0, 0], [0, 17, -17, 9, 0, 0, 7, 9, 7, -6], [0, -2, 2, 0, 0, 0, 15, 0, 0, -9], [-7, 0, -4, 17, 1, -6, 0, 0, 0, 0], [8, 0, -2, 8, -4, -13, 0, 0, 0, 0], [-8, -7, 0, 0, 2, 9, 8, -13, -9, 2], [6, -10, 0, 0, 6, 3, -6, -8, -3, 6], [10, -9, 0, 0, 2, -3, -10, -1, 3, 2], [4, -6, 0, 0, 7, -3, -4, 5, 3, 7], [0, -9, 9, 1, 0, 0, 17, 1, -13, -1], [-4, -12, 0, 0, 3, 12, 4, 0, -12, 3], [0, -5, 5, 14, 0, 0, -17, 14, 8, -15], [0, 0, 0, -3, 0, 0, 4, -3, 1, -2], [0, -15, 15, -4, 0, 0, 4, -4, -4, -2], [1, 12, 0, 0, -5, 1, -1, 2, -1, -5], [20, 0, 2, -2, 0, 8, 0, 0, 0, 0], [5, 5, 0, 0, -16, -8, -5, 18, 8, -16], [0, -9, 9, -7, 0, 0, 18, -7, -15, -7], [-3, 0, -22, 12, -3, 3, 0, 0, 0, 0], [9, 12, 0, 0, -4, -19, -9, 11, 19, -4], [0, 6, -6, -11, 0, 0, 11, -11, -19, 9], [6, 0, 10, -5, 2, 13, 0, 0, 0, 0], [-15, 0, 1, -5, 3, -1, 0, 0, 0, 0], [-2, 0, 14, -3, 2, -5, 0, 0, 0, 0], [-7, 0, -3, -2, 0, -7, 0, 0, 0, 0], [0, 6, -6, 6, 0, 0, 24, 6, -2, -1], [-31, -13, 0, 0, 11, 6, 31, -4, -6, 11], [0, 0, -2, -3, -9, -14, 0, 0, 0, 0], [9, -6, 0, 0, 6, 1, -9, 15, -1, 6], [-2, 0, 0, 0, -6, -5, 2, 7, 5, -6], [20, 0, 3, -7, 2, -7, 0, 0, 0, 0], [0, -1, 1, -11, 0, 0, 7, -11, 0, 2], [0, -2, 2, 9, 0, 0, -12, 9, -3, 9], [-11, 0, 8, -2, 2, 10, 0, 0, 0, 0], [-15, 0, -9, 3, -7, -5, 0, 0, 0, 0], [17, 8, 0, 0, -13, -8, -17, 14, 8, -13], [0, 0, 0, 5, 0, 0, 5, 5, 10, -19], [-12, 0, 2, -5, -11, 5, 0, 0, 0, 0], [0, -7, 7, -14, 0, 0, -4, -14, -16, 11], [12, 8, 0, 0, 17, 0, -12, 1, 0, 17], [0, 0, -18, 3, 1, 4, 0, 0, 0, 0], [13, 14, 0, 0, -9, -6, -13, -4, 6, -9], [0, -17, 17, -8, 0, 0, 6, -8, 3, -5]]
